Waveguide bend



Nov. 24, 1959 HANS-GEORG UNGER 2,914,741

WAVEGUIDE BEND 3 Sheets-Sheet 1 Filed Aug. 29, 1957 U 7' lL/Z ING MEANS TAPERED CURVATURE NORMAL MODE BEND CONS MN T CURl ATURE CIRCULAR ARC IN [/5 N TOR By H. a. UNGER /T h/ZQ Z A TTORNE Y Nov. 24,- 1959 Filed Aug. 29, 1957 cURmruRE A cuRmrukE-A HANS-GEORG UNGER 2,914,741

WAVEGUIDE BEND 3 Sheets-Sheet 2 FIG. 3

LENGTH LENGTH lNl/ENTOR /1'.G. UNGER ay /z A T TORNEV HANS-GEORG UNGER 2,914,741

WAVEGUIDE BEND Nov. 24, 1959 Filed Aug. 29, 1957 3 Sheets-Sheet 3 C URVA TURE 1A CURVATURE 36 LENGTH INVENTOR H. G. UNGER as azjL ATTORNEY 2,914,741 Patented Nov. 24,1959

WAVEGUIDE BEND Hans-Georg Unger,,Lincroft, assignor to Bell Telephone Laboratories, Incorporated, New York, N.Y., a

corporationv of New York, d r Application August 29, 1957, Serial No. 681,027 S Claims. 0. 333-981 This invention relates ,to guided electromagnetic wave transmission and, more particularly, to. waveguide configurations for transmitting the TE circular electric wave mode through curved waveguide sections. In this specification, the term bend will be considered generic to all-waveguide configurations which deviate smoothly froin'axial linearity.

ErAs is: now wellappreciated in, the wave transmission art, the propagation of microwave energy in the form of TE -waves, in circular waveguides is especially suited to long distance transmission since the attenuation characconversion losses sulfered by wave energyin the circular,

electric wave mode in waveguide bends. s It is a further object to define physical waveguide bend configurations which are attractive from an energy trans mission viewpoint.

A more specific. object is to bend an elastic waveguide section into a configuration, having a tapered curvature through the application of an external force at the enter of the section directed radially opposite to the external forces applied at the section extremities. i

In accordance with the invention, it, has been dis covered that ordinary prior art waveguide bends, which are of several general types, donot provide the lowest attainable conversion loss for circular electric wave transmission when utilized in conjunction with one of the modification techniques hereinabove noted. it has been further discovered that TE wave mode transmission losses in bends may be substantially reduced by tapering the curvature of the curved waveguide section from zero at its junction with the straight waveguide section to a given maximum value and subsequently tapering the teristic, of this mode, unlikethat ofrnost other modes, 1

decreases with increasing frequency. However, a major diificulty arises in conjunction with this mode of transmission from the fact that the T13 mode is not the dominant transmission mode in the hollow pipe type wave ,guide ofcircular.crosssection. 1. Thus energy maybe lost to. other wave modes capableof transmission within the guide. *Propagation of TE waves in an ideal round waveguide which is perfectly straight, uniform, andgconducting is essentially undisturbed. However, slight imcurvature from this given. value to Zero. Such a wave guide curvature characteristic provides a-smooth physical and electrical transition at the junction between the angularly related straight waveguide sections and the curved waveguide member joining them. A waveguidebend in accordance with the invention has a radius of curvature associated therewith at a point just beyond the junction considerably greater than that which would normally re suit, from the stresses and 7 strains associated with an ordinary bent membero; Such a novel waveguide bend is designated atapered curvaturenormal mode bend. As

i will be more fully explained in a later portion of this perfections in the guide itself and,-more.especially, any 31 deviation from straightness of thelongitudinal axisof the waveguide may excite waves of other modes and thus produce serious losses. These lossesare due-mainly to the fact thagwaveguide curvature induces acoupling between the desiredjTEi and other wave transmission modes. The

majorconversion problem is that involving the IE and i the TM modes. To a much less serious extent, losses occu'rE-between the TE and TE modes as well as between the TE and other modes of higher 'order. The ease with which conversion to TM occurs is explained by an. equality, in perfectly conducting uniform round waveguide, between the propa ation constants of the TE andfthe TM modes.

t 7 Various structural modificationtechniques have been suggested for'improving c'ircular electricwavepropaga tion through circular waveguides. For example, some of these structures include a circularly corrugated waveguide, as disclosed in United States Patent 2,751,561, which issued June 19,1956, to A. P. King; a slotted waveguide as disclosed in United States Patent 2,779,006, is-

. sued January 22, 1957, to W. I. Albersheim; a helix waveguide as disclosedin the .copending applications of J.' R. Pierce, Serial No. 416,315, and S. E. Miller, Serial No. 416,316, both filed March 15, 1954, now United States Patents 2,848,695 and 2,848,696,-respectively, both issued August 19, 19.58; or a dielectrically lined waveguide-as disclosed in applicants copending application Serial No. 681,054, filed'August29, 1957. Eachoffthe above-noted structures removes the TE -TM propagation constant degeneracyand thus reduces mode conver sion loss from the circular electric wave mode in curved sections of circular waveguide. In many waveguide applications, necessity for relativelysharp bends arises. In

such bends, mode conversion losses are undesirably high even in these modified structures J r It .is. therefore, an object of this invention to reduce specification, a tapered curvature normal mode bend is fabricated by bending a waveguide section while under the influence of at least one external force directed radially outward at. the point of maximum curvature. Physically, this indicates that a bend in accordance with .theinvention is formed by bending a waveguide section around at leastonefixed point. I

In accordance with a first principal embodiment of the invention, a waveguide section is physically shaped to possess a curvature whichtconforms with the graphical representation of theFourier transform of a Tscheby scheif polynomial of infinite degree. Such a configuraf tion represents an optimum tapered curvature normal mode waveguide bend from an electrical transmission viewpoint. I n r p In accordance with a second principal embodiment of the invention, a waveguide section is physically shaped to possess a tapered curvature characteristic which gradu ally.increases fromgero over a first portion of its length and .al tape redcurvature characteristic which gradually decreases tQI ZGIOQVQS' a second-portion of its length. These portions may be directly connecte'lor they may be connected by a third portion having constant curvature.

, transmission system;

" versus curvature characteristic Such. a configuration represents ,an optimum tapered curvaturenormalmode waveguide, bend from a mechanical or fabricational viewpoint. The above and other bjects, the natureof the present morefully upon consideration of the drawings and the detailed description thereof which follow, a 1

. In the drawings: 1

Fig. 1 is a perspective view of a simplified waveguide Fig. 2 illustrates a pair of angularly related straight waveguides connected by several types of waveguide bends; j

Fig. 3 is a graphical representation of the length of a prior art waveguide bend;

, outloss ,ex c'eptfor dissipation.

-Fig. 4'is a graphical representation of the length versus curvature characteristic of one waveguide embodiment of the invention;

Fig. 5 is a waveguide bend having the curvature characteristic 'of Fig. 4;

Figs. 6A and 6B are graphs illustrating the length versus curvature characteristic of additional waveguide embodiments of the invention; and

Fig. 7 illustrates a method of fabricating a curved waveguide section having the curvature characteristic of Fig.-6B.

7 Referring more specifically to the drawings, Fig. 1 illustrates a simplified microwave installation comprising microwave. source'10 supplying energy in the form of TE waves to a microwave utilizing means 11 through a continuous circular waveguiding passage. Utilizing means 1-1 maybe .a microwave amplifier, a receiver, or an antenna, for example. The circular waveguiding passage comprises angularly related straight sections 12, 13 which are joined by smoothly curved section 14, the paJticuIar design of which will be explained in detail in a later portion of this specification.

In a'microwavc system for the transmisison of TE waves, such as, for example, the system of Fig. 1, the inside radius a of the circular pipe guide selected for the propagation of these waves must be greater than the critical or cut-off radius 01 for the TE mode. For the TE mode, a is equal to 0.611 where n is the wavelength in free space of the lowest frequency wave in the transmission band. In practice, a is 'made greater than a and may vary in different systems from' 1.51 to 151 For illustrative As stated above, this mode conversion may be reduced by modifying section 14 in such a way as to remove the propagation constant equality between the TE and TM wave modes. However, mode conversion to TM wave energy is not completely eliminated by such modification. In addition, mode conversion to modes other than TM persists.

It is clear that a curved waveguide transition between angularly related straight waveguide sections which transforms the TE normal mode of thestraight waveguide into only one of the normal modes of the curved guide and, by the same token, transforms a particular normal mode of the curved waveguide into only the I TE normal mode of the straight guide will avoid all purposes, a suitable inner radius for the waveguide structures to be described hereinmay be 7. 7a or 4.7M. Thus if a hollow ripe guide two inches in diameter were I chosen fortransmission of the TE waves, t in accord- 'nect, both physically and electrically, angularly related straight waveguide sections 12 and 1 3. TE wave propagation'through curved seetion14 is most'easily explained in terms of normal modes; thatis, those waves for' a particular waveguiding structure which propagate with- Normal modes represent solutions of the wave' equations inthe particular waveguiding structure being investigated. In a straight waveguide excited by a source of TE wave energy, the normal mode of interest is the TEm mode; There are other normal modes capable of propagating in straight waveguide sections but, in the absence of waveguide discontinuities therein which would cause conversion to these other modes, they need not be considered. The

normal modes of a curved waveguide section are not so simple as those of a straight section. They may, however, be expressed as'a sum of the straight guide normal modes. In the present invention, the normal mode in the curved guide which is of'interest is that one which, when represented as a sum of straight guide modes, has the greatest part of its power in the TE mode portion of the sum. Thus the curved guide normal mode of interest is very similar to the T E mode of the straight guide.

I At a transition from a straight waveguide section to a curved waveguide section, such as between sections 12 and 14 of Fig. l, the normal mode of straight guide section 12 (TE will excite not only this normalmode but a series of other normal modes in curved section 14. This combination of normal modes propagates through section 14 and, at the transition between section 14 and straightsection 13 excites not only the TE modebut a series of other normal modes of the straight waveguide.

'The total power inmodes other than the TE mode in section 13 thus represents mode conversion loss introduced by curved section 14.

mode conversion losses. In accordancewith the invention, such a waveguide transition can be realized by tapering the curvature of the curved waveguide section from zero at its junction with the straight waveguide. In Fig. 1,'for example, if the curvature of bend 14 is tapered over its length from zero to a finite value and then back to zero, the normal TE' mode incident from straight section 12 will be gradually transformed into the particular normal mode of bend 14 which is most similar in field configuration to the circular electric wave. At each point along the taper, there is only one normal mode consistent with the value of curvature presented by the bend to the propagating wave energy as well as being similar in field configuration to the TE mode. By gradually tapering the curvature of the bend the propagating wave energy is retained in the desired normal wave mode throughout the bend. Such a bend is the tapered curvature normal 'mode bend.

It is clear that the waveguide sections under consideration in this specification have physical dimensions large enough to permitmore than one wave mode to propagate therein. In such a situation, the two or more wave modes which may propagate together have differing phase constants. This is true even for the TE -TM phase constants in the modified bend section 14 from which their inherent degeneracy has been removed. Since the phase constant of the TM as well as those of other modes capable of propagating in the circular waveguide are different from that of the TE there will be a periodic phasal reinforcement or'addition and a periodic phasal destruction or subtraction between the TE and each of the unwanted higherorder modes. Associated with each mode pair comprising the TE and oneof the unwanted higher order modes is a periodic distancebetween consecutive points of equal amplitude and phase. This distance is called the beat wave length, A f or'that particular mode pair. In order for the tapered curvature portion of the normal mode bend to function properly, its, length z should be of the order of or longer than the longest heat wave length associated with the structure. That is, since It can be seen that, for plain circular waveguide bends; that is, those in which the degeneracy between the TE and 'TM phase constant has not been removed, the normal mode. taper would have to be infinite in length. Thus, a nonde g'enerate waveguide becomes an essential condition for a normal mode bend in accordance with the present invention, V ,Fig. .2 illustrates in comparative relationship, the physical shape of an embodiment of the present invention together with physical shapes of the general types of P9ibl P r Wa eguide bends. In Fig. 2, segments 20,

' throughout.

21 i epresent angularly related straight waveguide sec;

tions to" be joined by a curved waveguide sectionfatl junctions 22 23. Sections 20, 21 are illustrated as being angularly related by 9.0 degrees'but the invention may 7 be utilizedto connect waveguide sections of any angular relationship. In prior art waveguide bends, a curved section of constant bending radius was thought to be. the ideal waveguide bend configuration for joining seg-- December 18, 1956, to S. E. Miller, bends'of constant" curvature, or constant bending radius, are utilized Such a bend is illustrated in 24 of circle 29 which has a radiusR I In the prior art, almost exclusive attention was directed tothe. problem of maintaining a" smooth change of dir'ection of the longitudinal axis of curvedwaveguide sections. What wasnot realized is that thecurvature associated with the. waveguide bend should also be changed smoothly, that is, a smooth change. in .the degree of curvature is also desirable. Thus, in Fig. 2, even though straight sections 20. 21 are tangent to are 24, thus providing a smooth change in direction, such an arc is of constant curvature while straight sections at junctions 22, 23. Such a discontinuity'will'cause the normalstraight guide TE wave mode,propagating in strai ht section for example, to' be transformed into a plurality of curved waveguide normal'rri'odes upon entering are 24 and into a plurality of straight waveguide normal modes upon exiting arc 24 andentering section 21. Such a transformation results in prohibitively hightransmission loss. i r

. Curved segments 25, 26 represent physicalshapesl'of curyed waveguide sections for Whichthe linear dimension of the.wave path is too short and too long, respectively, to form the constant curvature bend represented by arc;;24. Segment presents a curvature discontinuity at junctions 22,:23. even more severe than'that ofarc 24. .Both arc 24 and segment, 25 have a radius of, curvature associated therewith at a location just beyond their junction points with the straight sections considerably less than doesthe tapered curvature normal mode bend in accordance;with the present invention.

Curvedlsegment 26, presents," in addition to acurvature discontinuity at junctions 22, 23, a curvature characteristic which reversessense twice overits longitudinal extent. That is, as'curved segment 26,.is' traversed be-.

Fig. 2as arc directed with respect flte .thel icurvaturel atjits are; of

contact with the waveguide sec tion. s

The midpoint of this area of contact represents-a point of maximum curvature :of the bend. ;Circle 31,

. is tangent to straight waveguide sections 20, 21 at junc-f tions 22, 23. As will become more evident hereinafter,

tween angularly related straight' sections .20, 21,". proce'eding,;ifor example, in the direction from section20 to section .21, the rotational sense associated 'with, the curvature changes from counterclockwise to. clockwise and subsequently from clockwise back to counterclockwise.- 'A bend in accordance with-the invention is described by a direction of;curvature of constant sense.

iln Fig.- c'urved fsegrnent 27 represents thephysical shape of a tapered curvature normal mode bend in accordance with the invention. From.Fig. 2, it may be I seen qthat, given a pair of; angularly related straight waveguide sections, the length of a'ta'pered curvature bendor transition section joining them'is greater than the length of a bend of constant curvature. However, the tapered curvature transition is not one into which a waveguidesection will deform under the influence of the normalstresses and strains associatedi'with the application of bending moments "at the extremities of the sections. In order. that a tapered} curvature bend in accordance with the invention he produced, it is essential, that at least one fixed point apart from the section extremities .be fixed. V Such a fixed point is represented by element 28 in Fig. 2. Fixed point or support 28 providesan externally applied force to section 27 outwardly portions of the bend.

which has a radius R is seen to be tangent to segment 27 at point28. Since R R it is clear that the "tapered curvature bend has a radius of curvature at its center point considerably less than that of circular arc 24 which a tapered curvature waveguide bend may be formed about a single fixed point support, about a plurality of such supports, or about a single circular form of constant" radius.

which the present inventionis distinguished from random prior art waveguide bends. f i i s Fig. "3 illustrates. the lengthlversus curvature 'charac teristic of a circular arc of constant bending radius such as for example, arc24o fFig 2. At points 22, 23, representing junctions 22, 23 of Fig.2, the curvature suffers an abrupt discontinuity. It is this type of dis continuity and its-associatedmode conversion effects which the present invention eliminates.

Fig. 4 illustrates the lengt as afunction of its length as where .Mathe'matically, such. a curvature may be expressed l=='modified Bessel function, and

whereA/B is the' smallest difference in phase :eonstarits and l is the total axial length of the tapered curvatur e The curvature characteristic shown inFig. 4 as curve 30 is. substantially a graphical representationlof this Fourier transform. I I I v Fig. 5 shows a'physicalwayeguide embodimentof a tapered curvature bend having the optimum curvature characteristic of Fig. 4. Since.this curvaturecharacteristic is embodied .jin'a structure :which has, a physical shape different from those described above in connection with Fig. 2,;it is essential to. the fabricationofsuch a specialized bend configuration toutilize, a' physically constraining bending form. t w In Fig. 5, such a-form comprises rectangular sheet or slab 40 whichhas template 41,,and clampingsupports 42 disposed on its surface. Template 41. is designedto have, a; physical shape corresponding to the particular; bend. shape desired. In Fig.5, template 4Lisshaped to possess the curvature characteristic of Fig. 4. Waveguidesection 43 is placed over template 41 and physically deformed to conform to the shape thereof. Clamping supports '42 .are then fastened to slab 40 in order to preserve the Template 41 and supports 42 apply In all cases, the additional force or forces .,pro vided by these supports produces the novel curvature by V ments of optimum design from a mechanical or fabricational viewpoint. -Referring specifically to Fig. 6A, 'a wayeguide section described by the curvature characteristic represented thereon comprises two distinct regions. The first of these regions, represented by segment 50, is described by'a curvature which-tapers linearly from zero to a finite yalue k The second region, represented by segment 51, is the reverse of segment '50 and is describedfby a curvature which tapers linearly from value [c to zero, (The total .length of the curved section com- -prising. curvature segments 50, 51 is defined as l and each of the regions of tapered curvature has a length Z Referring now to Fig. 6B, a waveguide section described'by' the curvature characteristic represented thereon'cornprises three distinct regions. The first of these regions, represented by segment 52 is described by a curvature which tapers linearly from zero to a finite value 19,. The second region, represented by segment 53, is described by a constant curvature k over its length;

The-thirdregion, represented by segment 54, is the re- Y T$haracteristic of segment '52 and is described by aicurvature which tapers linearly from value k to'zero. The total length of the curved section is again defined as-l andeach of the regions of tapered curvature has a length. 1 v For a given bending angle, that is the angle through which the direction of the waveguide line is desired 'to be changed, an optimum total length I may be derived. The derivation of such an optimum relationship will be presented in a later portion of this specification;

The curvature characteristic of Fig. 6B is realized in the waveguide embodimentillustrated in Fig. 7. In Fig. 7, waveguide section "60" comprises a conductively bounded hollow round pipefrom which the T E 'TM phase constant degeneracy has been removed. This pipe is bent around circular form 61 in a manner not exceeding the elastic deformation limits of the physical waveguide.

Form 61 is illustrated as being circular. The

important property offonn 61 is that it has a constant curvature--over its area of contact with waveguide 60. The radius R of form 61 thus becomes the minimum bending radius of waveguide section-60.,v Bending is preferably accomplished by applying forces F, indicated by arrows 62, 63 at the extremities of waveguide section 60s These inwardly directed forces, which are caused to act upon both ends of section 60 produce, with form 61 ,,a torque which: bends section 60 in a manner producing the curvature versus length characteristic of Fig. 6B..

Essentially, form 61 provides the outwardly directed force noted above which imparts the tapered curvature characteristic to section -60. ThllS,fIOI1l the left end of-section 60 to the point of contact 64 between guide 60 and form 61, a distance 31, the pipe curvature increases linearly fromzero (descriptive of straight waveguide) to the constant value-associated with form 61 over its area'of contact with-section 60. Between points 64-and 65 waveguide 60 and form 61 are in mutual contact. Thus the curvature of waveguide 60 has a con stant valueover this interval. From point of contact 65 to-the right end ofsec tion 60, again a distance 2 the curvature decreases linearly from the constant value as-' sociated withform '60 to zero. For an embodiment of -the invention in which the constant curvature of form 61 is defined as k and the point of contact between form 61 and section 60 along the length l of said section is defined as Z the curvature k at any point along the lineartaperis V I shown for Fig. 6B. .That is, in Fig. 7, the region of contact of guide 60 and form 61 is reducedsubstantially to zero. of "the distributed force provided byf orm 61. tice this could be accomplished by bending a waveguide section over a fixed pin having a radius negligiblewith respect to the'length of the waveguide bend. Bending would be accomplishedthrough the application of forces, identical to forces 62, 63 of Fig. 7, at the extremities of the waveguide section to be bent.

Mathematically, a curve described by the curvature expressed in (3) is known as Cornus spiral. As already noted, the length of the tapered curvature sections in a waveguide embodiment of the invention must be of the order of or greater than the largest beat wave length associated with the curved waveguide section. V

Practically, there is ,a minimum total length of the curved section imposed by the requirement that the bending may not. exceed the elastic deformation limits of the guide for the fabrication method shown in Fig.7. That is, for a given pipe, the shortest allowable bending radius R for form 61 is rain where f =flexural stress at elastic limit =modulus of elasticity of pipe r =outside radius of pipe For a given specified angle 0 through which the direction of a waveguide line is desired to be changed, the minimum bending radius defined in (4) above requires a minimum total length of waveguide 1 of min= 0 m1n 'to the invention, that is, outwardly directed at the center of the curved member and inwardly directed at its extremities, would remain;

When a constraining form is used and the elastic limit of the waveguide is exceeded, the bending form may be removed after bending is completed. However, there will be a certain fspringback factor associated with the metal pipe and this must be taken into account when the layout of the constraints is planned.

, Asabove noted, it has been found that, for a given bending angle 0 a bend geometry which minimizes total bend loss can be determined. In general, the total bend loss A in a normal mode bend can be expressed as 1 p 1-4u A c1z+C1 W (6) in which C C and C are quantities depending on total bending angle, physical waveguide parameters and frequency; and

Since the quantities C C and C depend upon the physical parameters of the waveguide bend it is evident that they are in part dependent upon the choice of a particular one of the above-mentioned techniques'for removing the TEQ-"l'Mi phase constant degeneracy'in plain A pointforce is outwardly applied in place In prac mode bend where V =attenuation constant of the TE wave mode in plam waveguide, a =attenuation constant of the TE; or TM; wave mode in plain waveguide, I e=e'-]'e", the real and imaginary parts respectively of the relative dielectric permittivity e of the dielectric liner, p ;=3.83170,

the cut off factor in a plain waveguide of radius a,

relative thickness of dielectric coat,

p =plain waveguide phase constant of TE waves,

Afl=ditference in phase constant between the TE and any of the coupled modes, and

c'=factor in the curvature coupling coetficient c=ck whichdepends on physical waveguide parameters and frequency.

The summation signs in Equations 7 indicate that all con-- pled modes should be taken into account.

The, necessary conditions for A( u, l) to be a minimum are arms" "3 rr=1 (8) Sufficient conditions for A(u, l) to be a minimum are To find the optimum bend geometry for a given dielectrically lined normal mode bend having a specified bending angle, y is first calculated from (12). If y l, the

I optimum geometry is where w is positiveiroot the calculated y. If y l, the optimum geometry is The normal mode bend is inherently a broad band device. Some terms contributing to total bend loss decrease with frequency while others increase. The over-alll frequency dependence, other than the oscillations of the mode conversion portion of the total loss as caused by spurious mode'ph'asing, is vof the same order as is the frequency dependence of the loss in the straight waveguide.

In all cases, it is understood that the above-described arrangements are illustrative of only some of the many possible specific embodiments that represent application of the principles of the invention. Numerous and varied other arrangements can readily be devised in accordance with these principles by those skilled in the art without departing from the spirit and scope of the invention.

What is claimed is:

l. A tapered curvature normal mode waveguide bend having first and second terminal ends for transmitting the circular electric wave mode between angularly re-- lated waveguide paths, said bend comprising a sectiom of bounded waveguide which is curved with a constant sense as the bend is traversed and in which the TE and. TM wave modes have difierent propagation constants, the tapered curvature portions of said section each having a length greater than the longest beat wavelength: associated with said section and each tapering between zero curvature at said terminal ends and a given curva' 2. A transmission system having first and second terminal ends for electromagnetic wave energy in the circular electric wave mode, a source of said wave energy connected to said first terminal end, utilizing means for said energy connected 'to said second terminal end, and at least one curved section of nondegenerate circular waveguide between said source and said utilizing means, said section having at least one region of curvature taper: ing smoothly in one sense from zero over a length greater than the longest beat wavelength, associated with said section.

3. A transmission system for electromagnetic wave energy in the circular electric wave mode comprising first and second angularly related straight waveguide sections joined ,by a smoothly curved waveguide section which is curved in one sense between said straight sections, said curved section having a radius of curvature at a locationjust beyond the junction with said first straight section considerably greater than that associated with a circular arc mutually tangent to said straight sections, said curved section having a radius of curvature at the center thereof considerably less than that associated with said arc, said curved section having a radius of curvature at a location just before the junction with said second straight section considerably greater than that associated with said arc, said curved section having a length between said junction and said center greater than the longest beat wavelength associated with said section. a j i 4. The waveguide bend according to claim 1 in which said. tapered curvature portions are joined by a section of v waveguide having said given curvature over its length.

5. The waveguide bend according to claim 1 in which said tapered curvature portions have a physical contour whose curvature is described by the Fourier transform of a Tschebyschetf polynomial of infinite degree.

(References on following page) .11 V Refe rgnces Cited inthe filq of ,this patent I v 7 OTHER REFERENCES 4 "'UNITEDST-ATE'S PATENTS y Cl linj-Flh *o t'i um 'Taper e d Trgnsmiss io n Li e 2,649,578 -in 18, 1953 Matching Section in Proceedings of the IRE, April 1956, 2,774,945 Miller Dec. 18, 1956 5 i ffi 2gi th IRE J 1956 0 use ce 1 FOREIGN PATENTS p 9: 3. o e anuary pages 31 35. 732,443 Great Britain June 22, 1955 

